**Added 10/25/06**

Cahill's Extra Quantum Gravity Term

*My comments are in italics*

*Reginald Cahill has come up with a new derivation for the
way gravity acts [1,2], based on looking at a generalized vector
velocity field, which gives an extra term proportional to
the fine structure constant alpha. This connects his theory to quantum
effects through Planck's constant, and gives some rationale for
the experimental fact that gravity acts almost instantaneously rather
than propagating at light speed. But the biggest effect of the new
term is to create conditions under which gravitational attraction can
vary somewhere between 1-over-r-squared and 1-over-r under
certain conditions. This provides a new explanation for: 1)
the apparent decrease in gravitational acceleration in deep (a
few km) bore holes in arctic ice; 2) an explanation for the apparent
mass of the central black hole in various types of galaxies, and
3) an explanation for the rotation curves of spiral galaxies
without resorting to theoretical dark matter. Cahill refers to all of
these as black hole effects, while I will refer to them as extra
mass effects.*

Cahill begins: Our understanding of gravity is based on Newton's
modeling of Kepler's phenomenological laws for the motion of the
planets within the solar system. In this model Newton took the
gravitational **acceleration field** to be the fundamental
dynamical degree of freedom, which is determined by the matter
distribution; essentially via the "universal inverse square
law", *i.e. Newton began with inverse square, so deviations to
this would not be apparent.*

If, rather than using an acceleration field, a** vector velocity field**
is assumed to be fundamental to gravity, then we immediately find
that "*extra mass*" effects arise as a space
self-interaction dynamical effect, and that the observed correlation
is simply that Mextra/M = alpha/2 for spherical systems, where alpha
is the fine structure constant, 1/137.036, *which contains Planck's
constant giving gravity a quantum component. The extra mass is only
0.365% of the actual mass, so it would not ordinarily be
observable except that it produces quite different dynamical behavior
in some situations!*

Let us investigate in a phenomenological way the consequences of
using a vector velocity field **v**(**r**, t) to be the
fundamental dynamical degree of freedom to model gravity. This form
is mandated by Galilean covariance under change of observer. In terms
of the velocity field, Newtonian gravity dynamics involves using **r**-dot
to construct a rank-0 tensor that can be related to the matter
density rho. The coefficient turns out to be the Newtonian
gravitational constant G. This is clearly equivalent to the
differential form of Newtonian gravity outside of a spherical mass M
which gives the usual inverse square law

The simplest non-Newtonian dynamics involves the two rank-0 tensors
constructed at 2nd order from partial(vi)/partial(xj). Hence the
modeling of gravity now involves two gravitational constants G and **alpha**,
with alpha being the strength of the self-interaction dynamics, but
which was not apparent in the solar system dynamics. All the various
experimental phenomena discussed herein imply that **alpha is the
fine structure** **constant **1/137 up to experimental errors.

When the matter density is **confined** to a sphere of radius R we
find that the "*extra mass*" density is confined to
that sphere, and that consequently g (r) has an inverse square law
behavior outside of the sphere. We find inside radius R that the
total "*extra mass*", to Order (alpha), is Mextra = M
(alpha/2), independently of the matter density profile. This turns
out to be a very useful property as knowledge of the density profile
is then not required in order to analyze observational data.

**Galaxies in General**

*The data for a number of galaxies is given in the paper, with the assumption
that the extra mass is all in the central black hole.*

*These data, when plotted on a log-log scale, cluster around
alpha/2, with deviations that depend on galaxy type that shows that
some of the assumptions in the theory are only approximate in these cases.*

Mextra/M, in particular, for globular clusters M15 and G1 and
highly spherical "elliptical" galaxies M32, M87 and NGC
4374, show that this ratio lies close to the "alpha/2-line",
where alpha is the fine structure constant 1/137. However for the *presumably
dynamically varying* spiral galaxies their Mextra/M values do not
cluster close to the alpha/2-line. Hence it is suggested that these
spherical systems manifest minimal "black hole" dynamics.
However these dynamics are universal, so that any spherical system
must induce such a minimal black hole mode, but for which, **outside
of such a system, only the Newtonian inverse square law would be apparent.**

**Bore Holes**

This mode must also apply to the Earth, which is certainly a
surprising prediction. However just such an effect has been
manifested in measurements of g in mine shafts and bore holes since
the 1980's. Data from these geophysical measurements give us a very
accurate determination of the value of alpha. To understand this
bore-hole anomaly we need to compute the expression for g (r) just
beneath and just above the surface of the Earth. To lowest order in
alpha, this gives Newton's "inverse square law" for r >
R, but in which we see that the *effective* Newtonian
gravitational constant is GN = (1+ alpha/2 )G, which is *slightly*
different from the fundamental gravitational constant G. This is
caused by the additional "*extra mass*". Inside the
Earth *a few kilometers* we see a g (r) different from Newtonian
gravity. The effect is that g decreases more slowly with depth than
predicted by Newtonian gravity. Delta g (r) is found to be, to 1st
order, *linear* in R - r. Delta g (r) = -2 pi alpha GN rho(R)(R
- r) for r < R, and Delta g (r) = 0 for r > R.

**Spiral Galaxy Rotation**

We now consider the *somewhat sparse* *density* situation
in which matter in-falls around an existing primordial black hole. *There
are measurements of the masses of various galaxies and their central
black holes.* Immediately we see some of the consequences of this
time evolution: (1) the acceleration field falls off much slower than
the Newtonian inverse square law, as this *local* in-fall would
happen very rapidly; (2) the resultant in-flow would result in the
matter rotating much more rapidly than would be predicted by
Newtonian gravity; and (3) this would result in a spiral galaxy
exhibiting non-Keplerian rotation of stars and gas clouds, via the "*extra
mass*" effect.

We can determine the star orbital speeds for highly non-spherical
galaxies in the asymptotic region by solving asymptotically where rho
is approximately 0. The velocity field will be approximately
spherically symmetric and radial; nearer in we would match such a
solution to numerically determined solutions. We then compute
circular orbital speeds giving the predicted "universal
rotation-speed curve". Because of the alpha dependent part, this
rotation-speed curve falls off extremely slowly with r, as is indeed
observed for spiral galaxies. *Thus, the essentially flat galaxy
rotation is explained by an extra gravitation effect rather
than by hypothetical dark matter. This also corresponds to Van
Flandern's [3] assertion that such rotation could be explained
if gravity varied as 1-over-r over the volume of the galaxy.* This
is illustrated in Fig. 3 for the spiral galaxy NGC 3198.

**Conclusion**

*If gravity is not always 1-over-r-squared, and has a quantum
component, then several previously unexplained phenomena can be
explained without resorting to dark matter. This becomes another good
argument against the validity of the Big Bang.*

**References**

1) R. T. Cahill, "Black Holes in Elliptical and Spiral Galaxies
and in Globular Clusters", **Progress in Physics, 3**, pp
51-56, October 2005.

2) R. T. Cahill, "Black Holes and Quantum Theory: The Fine
Structure Constant Connection", **Progress in Physics, 4**,
pp 44-50, October 2006.

3) T. Van Flandern, **Dark Matter, Missing Planets, & New Comets**,
North Atlantic Books, Berkeley, California, 1998.